Properties

Label 304.4.a
Level $304$
Weight $4$
Character orbit 304.a
Rep. character $\chi_{304}(1,\cdot)$
Character field $\Q$
Dimension $27$
Newform subspaces $11$
Sturm bound $160$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 304.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(160\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(304))\).

Total New Old
Modular forms 126 27 99
Cusp forms 114 27 87
Eisenstein series 12 0 12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(19\)FrickeDim
\(+\)\(+\)$+$\(8\)
\(+\)\(-\)$-$\(5\)
\(-\)\(+\)$-$\(7\)
\(-\)\(-\)$+$\(7\)
Plus space\(+\)\(15\)
Minus space\(-\)\(12\)

Trace form

\( 27 q + 2 q^{5} + 14 q^{7} + 243 q^{9} + O(q^{10}) \) \( 27 q + 2 q^{5} + 14 q^{7} + 243 q^{9} - 86 q^{11} - 46 q^{13} + 12 q^{15} - 26 q^{17} - 57 q^{19} + 136 q^{21} - 68 q^{23} + 865 q^{25} + 576 q^{27} - 398 q^{29} - 24 q^{33} - 246 q^{35} + 322 q^{37} - 540 q^{39} + 118 q^{41} - 230 q^{43} - 350 q^{45} + 1262 q^{47} + 915 q^{49} + 1012 q^{51} + 90 q^{53} + 830 q^{55} - 272 q^{59} - 390 q^{61} + 610 q^{63} - 348 q^{65} + 1020 q^{67} + 24 q^{69} + 1124 q^{71} - 1354 q^{73} + 1068 q^{75} + 1608 q^{77} + 1080 q^{79} + 1571 q^{81} - 1676 q^{83} - 1588 q^{85} - 3540 q^{87} - 314 q^{89} + 952 q^{91} - 2248 q^{93} + 760 q^{95} - 290 q^{97} + 1154 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(304))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 19
304.4.a.a 304.a 1.a $1$ $17.937$ \(\Q\) None \(0\) \(2\) \(-9\) \(31\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-9q^{5}+31q^{7}-23q^{9}-57q^{11}+\cdots\)
304.4.a.b 304.a 1.a $1$ $17.937$ \(\Q\) None \(0\) \(5\) \(-12\) \(-11\) $-$ $+$ $\mathrm{SU}(2)$ \(q+5q^{3}-12q^{5}-11q^{7}-2q^{9}+54q^{11}+\cdots\)
304.4.a.c 304.a 1.a $2$ $17.937$ \(\Q(\sqrt{73}) \) None \(0\) \(-9\) \(-9\) \(18\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-4-\beta )q^{3}+(-3-3\beta )q^{5}+(7+4\beta )q^{7}+\cdots\)
304.4.a.d 304.a 1.a $2$ $17.937$ \(\Q(\sqrt{177}) \) None \(0\) \(-1\) \(10\) \(-57\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(6-2\beta )q^{5}+(-28-\beta )q^{7}+\cdots\)
304.4.a.e 304.a 1.a $2$ $17.937$ \(\Q(\sqrt{57}) \) None \(0\) \(1\) \(-5\) \(6\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(-3+\beta )q^{5}+(5-4\beta )q^{7}+\cdots\)
304.4.a.f 304.a 1.a $2$ $17.937$ \(\Q(\sqrt{33}) \) None \(0\) \(5\) \(-5\) \(30\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(3-\beta )q^{3}-5\beta q^{5}+(17-4\beta )q^{7}+\cdots\)
304.4.a.g 304.a 1.a $3$ $17.937$ 3.3.7057.1 None \(0\) \(-4\) \(7\) \(-7\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{3}+(2+2\beta _{1}+\beta _{2})q^{5}+\cdots\)
304.4.a.h 304.a 1.a $3$ $17.937$ 3.3.35529.1 None \(0\) \(-1\) \(9\) \(-44\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}-\beta _{2})q^{3}+(3+\beta _{2})q^{5}+(-15+\cdots)q^{7}+\cdots\)
304.4.a.i 304.a 1.a $3$ $17.937$ 3.3.3144.1 None \(0\) \(-1\) \(14\) \(35\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+(4-\beta _{1}+\beta _{2})q^{5}+(11-2\beta _{1}+\cdots)q^{7}+\cdots\)
304.4.a.j 304.a 1.a $3$ $17.937$ 3.3.3221.1 None \(0\) \(5\) \(2\) \(35\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(2+\beta _{2})q^{3}+(1-\beta _{1}+2\beta _{2})q^{5}+(11+\cdots)q^{7}+\cdots\)
304.4.a.k 304.a 1.a $5$ $17.937$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-2\) \(0\) \(-22\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+(\beta _{1}+\beta _{2})q^{5}+(-4-\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(304))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(304)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 2}\)